### Abstract

An extension B⊂A of algebras over a commutative ringk is an H-extension for an L-bialgebroid H if A is an H-comodule algebra and B is the subalgebra of its coinvariants. It is H-Galois if the canonical map A ⊗_{B}A →A ⊗_{L}H is an isomorphism or, equivalently, if the canonical coring (A ⊗_{L}H:A) is a Galois coring. In the case of Hopf algebroid H=(H_{L}, H_{R}, S) any H_{R}-extension is shown to be also an H_{L}-extension. If the antipode is bijective then also the notions of H_{R}-Galois extensions and of H_{L}-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroid H with bijective antipode. It states that any H-Galois extension B⊂A is projective, and if A is k-flat then already the surjectivity of the canonical map implies the Galois property. The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra, extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery.

Original language | English |
---|---|

Pages (from-to) | 233-262 |

Number of pages | 30 |

Journal | Annali dell'Universita di Ferrara |

Volume | 51 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2005 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Annali dell'Universita di Ferrara*,

*51*(1), 233-262. https://doi.org/10.1007/BF02824833

**Galois theory for Hopf algebroids.** / Böhm, G.

Research output: Contribution to journal › Article

*Annali dell'Universita di Ferrara*, vol. 51, no. 1, pp. 233-262. https://doi.org/10.1007/BF02824833

}

TY - JOUR

T1 - Galois theory for Hopf algebroids

AU - Böhm, G.

PY - 2005

Y1 - 2005

N2 - An extension B⊂A of algebras over a commutative ringk is an H-extension for an L-bialgebroid H if A is an H-comodule algebra and B is the subalgebra of its coinvariants. It is H-Galois if the canonical map A ⊗BA →A ⊗LH is an isomorphism or, equivalently, if the canonical coring (A ⊗LH:A) is a Galois coring. In the case of Hopf algebroid H=(HL, HR, S) any HR-extension is shown to be also an HL-extension. If the antipode is bijective then also the notions of HR-Galois extensions and of HL-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroid H with bijective antipode. It states that any H-Galois extension B⊂A is projective, and if A is k-flat then already the surjectivity of the canonical map implies the Galois property. The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra, extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery.

AB - An extension B⊂A of algebras over a commutative ringk is an H-extension for an L-bialgebroid H if A is an H-comodule algebra and B is the subalgebra of its coinvariants. It is H-Galois if the canonical map A ⊗BA →A ⊗LH is an isomorphism or, equivalently, if the canonical coring (A ⊗LH:A) is a Galois coring. In the case of Hopf algebroid H=(HL, HR, S) any HR-extension is shown to be also an HL-extension. If the antipode is bijective then also the notions of HR-Galois extensions and of HL-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroid H with bijective antipode. It states that any H-Galois extension B⊂A is projective, and if A is k-flat then already the surjectivity of the canonical map implies the Galois property. The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra, extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery.

UR - http://www.scopus.com/inward/record.url?scp=33645789698&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645789698&partnerID=8YFLogxK

U2 - 10.1007/BF02824833

DO - 10.1007/BF02824833

M3 - Article

AN - SCOPUS:33645789698

VL - 51

SP - 233

EP - 262

JO - Annali dell'Universita di Ferrara

JF - Annali dell'Universita di Ferrara

SN - 0430-3202

IS - 1

ER -